Method of two-stage FRM filter

ABSTRACT

An improved design method of a two-stage FRM filter includes the following steps: constructing an improved two-stage FRM filter; calculating passband and stopband edge parameters of a prototype filter, passband and stopband edge parameters of a second-stage masking filter and passband and stopband edge parameters of a first-stage masking filter in Case A and Case B, respectively; calculating the complexity of the FRM filter according to the obtained parameters, and finding out one or more sets [M, P, Q] having the lowest complexity within a search range; and optimizing the improved FRM filter. The improved design method of a two-stage FRM filter has the following beneficial effect: as compared to a conventional design method of a two-stage FRM filter, the complexity of a narrow-band FIR (Finite Impulse Response) filter can be reduced through design using the improved method, and power consumption is thus reduced in hardware implementation.

FIELD OF THE INVENTION

The present invention relates to an improved design method of atwo-stage FRM filter.

BACKGROUND OF THE INVENTION

Frequency response masking (FRM) is an efficient method for designing anFIR (Finite Impulse Response) filter having a narrow transition bandcharacteristic. When a filter has a narrow transition band, multi-stageFRM may be utilized to further reduce its complexity.

A filter is composed of a prototype filter H_(a)(z) and two maskingfilters H_(ma)(z), H_(mc)(z). The transition band of the filter isprovided by an interpolation filter H_(a)(z^(M)) or its complement

$\left( {z^{- \frac{M{({N_{a} - 1})}}{2}} - {H_{a}\left( z^{M} \right)}} \right)$where M is an interpolation factor for H_(a)(z). The purpose of usingtwo masking filters H_(ma)(z) and H_(mc)(z) is to remove unnecessaryperiodic sub-bands.

Two-stage FRM filters are widely used in practice. The structure of atwo-stage FRM filter is as shown in FIG. 2. A transition band shapingfilter is represented by G(z). A constraint condition for interpolationfactors M, P, Q is as follows:M=kP=kQ  (3).

At present, various improvements made to the two-stage FRM filters arebased on the assumption of the equation (3). The satisfaction of theequation (3) leads to a direct problem that the second-stage output mustbe a periodic amplitude response. However, the second-stage output doesnot have to be periodic as long as it provides a desired transition bandfor a target filter. Therefore, the complexity of an FRM filter can befurther reduced if the values of the three factors can be found in awider range.

SUMMARY OF THE INVENTION

The present invention aims at solving the above problem and provides animproved design method of a two-stage FRM filter.

In order to achieve the above objective, the present invention employsthe following technical scheme.

An improved design method of a two-stage FRM filter comprises thefollowing steps:

(1) constructing an improved two-stage FRM filter having a transferfunction H(z) as follows:H(z)=G(z)H _(ma) ⁽¹⁾(z)+(1−G(z))H _(mc) ⁽¹⁾(z),wherein G(z)=H_(a) ⁽²⁾(z^(M))H_(ma) ⁽²⁾(z^(P))+(1−H_(a)⁽²⁾(z^(M)))H_(mc) ⁽²⁾(z^(Q)), without any constraint among interpolationfactors M, P, Q;

H_(a) ⁽²⁾(z^(M)) represents a prototype filter, while H_(ma) ⁽¹⁾(z) andH_(mc) ⁽¹⁾(z) represent first-stage masking filters, respectively, andH_(ma) ⁽²⁾(z^(P)) and H_(mc) ⁽²⁾(z^(Q)) represent second-stage maskingfilters, respectively;

(2) searching within a search range for [M, P, Q], and for a certain set[M, P, Q], calculating passband and stopband edge parameters of theprototype filter H_(a) ⁽²⁾(z), passband and stopband edge parameters ofthe second-stage masking filters and passband and stopband edgeparameters of the first-stage masking filters in Case A and Case B,respectively, on the basis that a transition band of the whole filter isprovided by H_(a) ⁽²⁾(z^(M)) or a complement of H_(a) ⁽²⁾(z^(M)); andcalculating the complexity of the FRM filter according to the obtainedparameters, and finding out one or more sets [M, P, Q] having the lowestcomplexity within the search range;

(3) optimizing the improved FRM filter according to the calculatedfilter parameters. A way of calculating the passband edge θ_(a) and thestopband edge φ_(a) of the prototype filter H_(a) ⁽²⁾(z^(M)) in the step(2) is as follows:

when the transition band of the whole filter is H_(a) ⁽²⁾(z^(M)), i.e.,in the Case A:m=└ω _(p) M/2π┘,θ_(a)=ω_(p) M−2mπ,φ_(a)=ω_(s) M−2mπ;when the transition band of the whole filter is the complement of H_(a)⁽²⁾(z^(M)), i.e., in the Case B:m=┌ω _(s) M/2π┐,θ_(a)=2mπ−ω _(s) M,φ_(a)=2mπ−ω _(p) M;wherein └x┘ represents a largest integer not more than x; ┌x┐ representsa smallest integer not less than x; whether the result satisfies0<θ_(a)<φ_(a)<π is determined in the two cases respectively, and if not,the results are abandoned.

Passbands of H_(a) ⁽²⁾(Mω) from 0 to π in the step (2) are orderlyrepresented by 0, 2, . . . ,

${2 \times \left\lfloor \frac{M}{2} \right\rfloor},$and passbands of 1−H_(a) ⁽²⁾(Mω) from 0 to it are orderly represented by1, 3, . . . ,

${2 \times \left\lfloor \frac{M - 1}{2} \right\rfloor};$assuming that the passband labeled as 2m of H_(a) ⁽²⁾(Mω) provides thetransition band, the passband of the masking filter from which thetransition band is extracted is defined as an effective passband; and inorder to reduce the complexity of the two first-stage masking filters,the following restrictive conditions are established:the passband 2m should be at least completely extracted; the passband2(m+1) should be completely fall outside an effective passband range.

When the transition band of the whole filter is provided by H_(a)⁽²⁾(z^(M)) i.e., in the Case A, a way of calculating the passband edgeω_(pma) ⁽²⁾ and the stopband edge ω_(sma) ⁽²⁾ of the second-stagemasking filter H_(ma) ⁽²⁾(z) is as follows:

(1) when masking is provided by H_(ma) ⁽²⁾(z^(P)), the case is denotedas Case_(p)=A:

$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {\omega_{1}P}},{{\omega_{2}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {{\omega_{3}P} - {2\pi\; p}}}\end{matrix};} \right.$

(2) when masking is provided by the complement of H_(ma) ⁽²⁾(z^(P)), thecase is denoted as Case_(p)=B:

$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {{2\pi\; p} - {\omega_{2}P}}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{1}P} - {2{\pi\left( {p - 1} \right)}}},{{2\pi\; p} - {\omega_{2}P}}} \right)}}\end{matrix};} \right.$wherein ω₁ is a left edge of the passband 2m of the prototype filter,while ω₂ is a right edge of the passband 2m of the prototype filter, andω₃ is a left edge of the passband 2(m+1) of the prototype filter; P isthe interpolation factor, which is a given value;p is an integer, and should satisfy the following condition:

${0 \leq p \leq \left\lfloor \frac{P}{2} \right\rfloor};$if no p satisfying the condition exists, the set of parameters isabandoned.

When the transition band of the whole filter is provided by H_(a)⁽²⁾(z^(M)), i.e., in the Case A, a way of calculating the passband edgeω_(pmc) ⁽²⁾ and the stopband edge ω_(smc) ⁽²⁾ of the second-stagemasking filter H_(mc) ⁽²⁾(z) is as follows:

(1) when masking is provided by H_(mc) ⁽²⁾(z^(Q)), the case is denotedas Case_(q)=A:

$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{4}P}},{{\omega_{5}P} - {2\pi\; q}}} \right)}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{6}Q} - {2\pi\; q}},{{2{\pi\left( {q + 1} \right)}} - {\omega_{7}Q}}} \right)}}\end{matrix};} \right.$

(2) when masking is provided by the complement of H_(mc) ⁽²⁾(z^(Q)), thecase is denoted as Case_(q)=B:

$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{6}Q}},{{\omega_{7}Q} - {2\pi\; q}}} \right)}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{4}Q} - {2\pi\;\left( {q - 1} \right)}},{{2\pi\; q} - {\omega_{5}Q}}} \right)}}\end{matrix};} \right.$wherein ω₄ is a left passband edge of the complementary filter passband2m−1 of the prototype filter; ω₅ is a right stopband edge of thecomplementary filter passband 2m−1 of the prototype filter; ω₆ is a leftpassband edge of the complementary filter passband 2m+1 of the prototypefilter; ω₇ is a right stopband edge of the complementary filter passband2m+1 of the prototype filter; q is an integer, and should satisfy thefollowing condition:

${0 \leq q \leq \left\lfloor \frac{q}{2} \right\rfloor};$if no q satisfying the condition exists, the set of parameters isabandoned.

When the transition band of the whole filter is provided by thecomplement of H_(z) ⁽²⁾(z^(M)), i.e., in the Case B, a way ofcalculating the passband edge ω_(pma) ⁽²⁾ and the stopband edge ω_(sma)⁽²⁾ of the second-stage masking filter H_(ma) ⁽²⁾(z) is as follows:

(1) when masking is provided by H_(ma) ⁽²⁾(z^(P)) the case is denoted asCase_(p)=A:

$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {P\;\omega_{4}}},{{\omega_{5}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{6}P} - {2\pi\; p}},{{2{\pi\left( {p + 1} \right)}} - {\omega_{7}P}}} \right)}}\end{matrix};} \right.$

(2) when masking is provided by the complement of H_(ma) ⁽²⁾(z^(P)) thecase is denoted as Case_(p)=B:

$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {\omega_{6}P}},{{\omega_{7}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{4}P} - {2\pi\;\left( {p - 1} \right)}},{{2\pi\; p} - {\omega_{5}P}}} \right)}}\end{matrix};} \right.$wherein ω₄ is a left passband edge of the passband 2(m−1) of theprototype filter; ω₅ is a right stopband edge of the passband 2(m−1) ofthe prototype filter; ω₆ is a left passband edge of the passband 2m ofthe prototype filter; ω₇ is a right stopband edge of the passband 2m ofthe prototype filter;p is an integer, and should satisfy the following condition:

${0 \leq p \leq \left\lfloor \frac{P}{2} \right\rfloor};$if no p satisfying the condition exists, the set of parameters isabandoned.

When the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B, a way ofcalculating the passband edge ω_(pmc) ⁽²⁾ and the stopband edge ω_(smc)⁽²⁾ of the second-stage masking filter H_(mc) ⁽²⁾(z) is as follows:

(1) when masking is provided by H_(mc) ⁽²⁾(z^(Q)), the case is denotedas Case_(q)=A:

$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{1}Q}},{{\omega_{2}Q} - {2\pi\; q}}} \right)}} \\{\omega_{sma}^{(2)} = {{\omega_{2}Q} - {2\pi\; q}}}\end{matrix};} \right.$

(2) when masking is provided by the complement of H_(mc) ⁽²⁾(z^(Q)), thecase is denoted as Case_(q)=B:

$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {{2\pi\; q} - {\omega_{2}Q}}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{1}Q} - {2{\pi\left( {q - 1} \right)}}},{{2\pi\; q} - {\omega_{2}Q}}} \right)}}\end{matrix};} \right.$wherein ω₁ is a left stopband edge of the complementary filter passband2m−1 of the prototype filter; ω₂ is a right passband edge of thecomplementary filter passband 2m−1 of the prototype filter; ω₃ is a leftstopband edge of the complementary filter passband 2m+1 of the prototypefilter;q is an integer, and should satisfy the following condition:

${0 \leq q \leq \left\lfloor \frac{q}{2} \right\rfloor};$if no q satisfying the condition exists, the set of parameters isabandoned.

When the transition band of the whole filter is provided by H_(a)⁽²⁾(z^(M)), i.e., in the Case A:

the passband edge ω_(pma) ⁽¹⁾ of the first-stage masking filter H_(ma)⁽¹⁾(z) is equal to ω_(p); a way of calculating the stopband edge ω_(sma)⁽¹⁾ of the first-stage masking filter H_(ma) ⁽¹⁾(z) is as follows:

ω_(sma)⁽¹⁾ = min (ω_(sma _ temp 1)⁽¹⁾, ω_(sma _ temp 2)⁽¹⁾), wherein$\omega_{{sma}\;\_\;{temp}\; 1}^{(1)} = \left\{ {\begin{matrix}t_{1} & {t_{1} \notin {R_{stop}(k)}} \\\omega_{8} & {t_{1} \in {R_{stop}\left( k_{1} \right)}}\end{matrix},{\omega_{{sma}\;\_\;{temp}\; 2}^{(1)} = \left\{ {\begin{matrix}t_{2} & {t_{2} \notin {R_{pass}(k)}} \\\omega_{9} & {t_{2} \in {R_{pass}\left( k_{2} \right)}}\end{matrix},{{\omega_{\theta} = {\left( {{2\pi\; k_{1}} = \varphi_{a}} \right)/M}};{\omega_{9} = {\left( {{2\pi\; k_{2}} - \theta_{a}} \right)/M}};{{R_{pass}(k)} = \left\lbrack {\frac{{2\pi\; k} - \theta_{a}}{M},\frac{{2\pi\; k} + \theta_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;{{R_{stop}(k)} = \left\lbrack {\frac{{2\pi\;\left( {k - 1} \right)} + \varphi_{a}}{M},\frac{{2\pi\; k} + \varphi_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;}} \right.}} \right.$

(1) when H_(ma) ⁽²⁾(z^(P)) is used for masking unnecessary band of anup-branch,t ₁=(2π(p+1)−ω_(sma) ⁽²⁾)/P;

(2) when the complement of H_(ma) ⁽²⁾(z^(P)) is used for masking theunnecessary band of the up-branch,t ₁=(2πp+ω _(pma) ⁽²⁾)/P;

(3) when H_(mc) ⁽²⁾(z^(Q)) is used for masking unnecessary band of adown-branch,t ₂=(2π(q+1)−ω_(smc) ⁽²⁾)/Q;

(4) when a down-branch complement of H_(mc) ⁽²⁾(z^(Q)) is used formasking unnecessary band,t ₂=(2πq+ω _(pmc) ⁽²⁾)/Q;wherein P, Q, M are interpolation factors; ω₈ is a right endpoint ofR_(stop)(k₁); k₁ is an integer satisfying t₁∈R_(stop)(k₁); ω₉ is a rightendpoint of R_(pass)(k₂); k₂ is an integer satisfying t₂∈R_(stop)(k₂);θ_(a) is a passband edge of H_(a) ⁽²⁾(z); H_(ma) ⁽¹⁾(z) φ_(a) is astopband edge of H_(a) ⁽²⁾(z).

When the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B:

the passband edge ω_(pma) ⁽¹⁾ of the first-stage masking filter H_(ma)⁽¹⁾(z) is equal to ω_(p);

a way of calculating the stopband edge ω_(sma) ⁽¹⁾ of the first-stagemasking filter H_(ma) ⁽¹⁾(z) is as follows:

ω_(sma)⁽¹⁾ = min (ω_(sma _ temp 1)⁽¹⁾, ω_(sma _ temp 2)⁽¹⁾), wherein$\omega_{{sma}\;\_\;{temp}\; 1}^{(1)} = \left\{ {\begin{matrix}t_{1} & {t_{1} \notin {R_{stop}(k)}} \\\omega_{8} & {t_{1} \in {R_{stop}\left( k_{1} \right)}}\end{matrix},{\omega_{{sma}\;\_\;{temp}\; 2}^{(1)} = \left\{ {\begin{matrix}t_{2} & {t_{2} \notin {R_{pass}(k)}} \\\omega_{9} & {t_{2} \in {R_{pass}\left( k_{2} \right)}}\end{matrix},{{\omega_{\theta} = {\left( {{2\pi\; k_{1}} = \theta_{a}} \right)/M}};{\omega_{9} = {\left( {{2\pi\; k_{2}} - \varphi_{a}} \right)/M}};{{R_{pass}(k)} = \left\lbrack {\frac{{2\pi\; k} - \theta_{a}}{M},\frac{{2\pi\; k} + \theta_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;{{R_{stop}(k)} = \left\lbrack {\frac{{2\pi\;\left( {k - 1} \right)} + \varphi_{a}}{M},\frac{{2\pi\; k} + \varphi_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;}} \right.}} \right.$

(1) when H_(mc) ⁽²⁾(z^(Q)) is used for masking unnecessary band,t ₁=(2π(q+1)−ω_(smc) ⁽²⁾)/Q;

(2) when the complement of H_(mc) ⁽²⁾(z^(Q)) is used for maskingunnecessary band,t ₁=(2πq+ω _(pmc) ⁽²⁾)/Q;

(3) when H_(ma) ⁽²⁾(z^(P)) is used for masking unnecessary band,t ₂=(2π(p+1)−ω_(sma) ⁽²⁾)/P;

(4) when the complement of H_(ma) ⁽²⁾(z^(P)) is used for maskingunnecessary band,t ₂=(2πp+ω _(pma) ⁽²⁾)/P;wherein ω₈ is a right endpoint of R_(stop)(k₁); k₁ is an integersatisfying t₁∈R_(stop)(k₁); ω₉ is a right endpoint of R_(pass)(k₂); k₂is an integer satisfying t₂∈R_(stop)(k₂); P, Q, M are interpolationfactors; θ_(a) is a passband edge of H_(a) ⁽²⁾(z); H_(ma) ⁽¹⁾(z)φ_(a) isa stopband edge of H_(a) ⁽²⁾(z).

(1) When the transistion band of the whole filter is provided by H_(a)⁽²⁾(z^(M)), i.e., in the Case A:

the stopband edge ω_(smc) ⁽¹⁾ of the first-stage masking filter H_(mc)⁽¹⁾(z) is equal to ω_(s);

a way of determining the passband edge ω_(pmc) ⁽¹⁾ of the first-stagemasking filter H_(mc) ⁽¹⁾(z) is as follows:

if t₃≥t₄, then

$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{3} & {t_{3} \notin {R_{stop}(k)}} \\{\max\left( {\frac{{2\pi\;\left( {k_{3} - 1} \right)} + \varphi_{a}}{M},t_{4}} \right)} & {t_{3} \in {R_{stop}\left( k_{3} \right)}}\end{matrix};} \right.$if t₃<t₄, then

$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{4} & {t_{4} \notin {R_{pass}(k)}} \\{\max\left( {\frac{{2\pi\; k_{4}} - \theta_{a}}{M},t_{3}} \right)} & {t_{4} \in {R_{pass}\left( k_{4} \right)}}\end{matrix};} \right.$

(a) when H_(ma) ⁽²⁾(z^(P)) is used for masking unnecessary band,t ₃=(2πp−ω _(pma) ⁽²⁾)/P;

(b) when the complement of H_(ma) ⁽²⁾(z^(P)) is used for masking theunnecessary band,t ₃=(2π(p−1)+ω_(sma) ⁽²⁾)/P;

(c) when H_(mc) ⁽²⁾(z^(Q)) is used for masking unnecessary band,t ₄=(2πq−ω _(pmc) ⁽²⁾)/Q;

(d) when the complement H_(mc) ⁽²⁾(z^(Q)) is used for maskingunnecessary band,t ₄=(2π(q−1)+ω_(smc) ⁽²⁾)/Q;

(2) when the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B:

the stopband edge ω_(smc) ⁽¹⁾ of the first-stage masking filter H_(mc)⁽¹⁾(z) is equal to ω_(s);

a way of determining the passband edge ω_(pmc) ⁽¹⁾ of the first-stagemasking filter H_(mc) ⁽¹⁾(z) is as follows:

if t₃≥t₄, then

$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{3} & {t_{3} \notin {R_{stop}(k)}} \\{\max\left( {\frac{{2{\pi\left( {k_{3} - 1} \right)}} + \varphi_{a}}{M},t_{4}} \right)} & {t_{3} \in {R_{stop}\left( k_{3} \right)}}\end{matrix};} \right.$if t₃<t₄, then

$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{4} & {t_{4} \notin {R_{pass}(k)}} \\{\max\left( {\frac{{2\pi\; k_{4}} - \theta_{a}}{M},t_{3}} \right)} & {t_{4} \in {R_{pass}\left( k_{4} \right)}}\end{matrix};} \right.$

(a) when H_(mc) ⁽²⁾(z^(Q)) is used for masking unnecessary band,t ₃=(2π(q−1)+ω_(smc) ⁽²⁾)/Q;

(b) when the complement of H_(mc) ⁽²⁾(z^(Q)) is used for masking theunnecessary band,t ₃=(2π(q−1)+ω_(smc) ⁽²⁾)/Q;

(c) when H_(ma) ⁽²⁾(z^(P)) is used for masking unnecessary band,t ₄=(2π(p−1)+ω_(sma) ⁽²⁾)/P;

(d) when the complement H_(ma) ⁽²⁾(z^(P)) is used for maskingunnecessary band,t ₄=(2π(p−1)+ω_(sma) ⁽²⁾)/P;wherein t₃ and t₄ are left passband edges of the passband including thetransition band and a first passband on the left of the transition band,respectively; k₃ is an integer satisfying t₃∈R_(stop)(k₃); k₄ is aninteger satisfying t₄∈R_(pass)(k₄); P, Q, M are interpolation factors;θ_(a) is a passband edge of H_(a) ⁽²⁾(z); H_(ma) ⁽¹⁾(z)φ_(a) is astopband edge of H_(a) ⁽²⁾(z).

The present invention has the following beneficial effects: according tothe present invention, the restriction that the interpolation factorsmust satisfy the constraint condition M=kP=kQ in a conventionalstructure is broken by constructing an improved two-stage FRM filterstructure, and the subfilters are optimized simultaneously by means of anonlinear joint optimization method. Results indicate that, as comparedto a conventional design method of a two-stage FRM filter, thecomplexity of a narrow-band FIR (Finite Impulse Response) filter can bereduced through design using the improved method, and power consumptionis thus reduced in hardware implementation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a structural schematic diagram of an improved two-stage FRMfilter of the present invention.

FIG. 2 is an amplitude response possibly existing in G(z) of the presentinvention.

FIG. 3 is an amplitude response of an up-branch of G(z) of the presentinvention.

FIG. 4 is an amplitude response of a down-branch of G(z) of the presentinvention.

FIG. 5 is a schematic diagram of calculation of a stopband edge ω_(sma)⁽¹⁾ in Case=A according to the present invention.

FIG. 6 is a schematic diagram of calculation of a passband edge ω_(pmc)⁽¹⁾ in Case=A according to the present invention.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The present invention will be further illustrated below in combinationwith the accompanying drawings and embodiment.

The structure of an improved two-stage FRM filter is as shown in FIG. 1in which M, P, Q are interpolation factors. A transfer function H(z) ofthe filter is expressed as the following formula:H(z)=G(z)H _(ma) ⁽¹⁾(z)+(1−G(z))H _(mc) ⁽¹⁾(z),  (4)G(z)=H _(a) ⁽²⁾(z ^(M))H _(ma) ⁽²⁾(z ^(P))+(1−H _(a) ⁽²⁾(z ^(M)))H _(mc)⁽²⁾(z ^(Q)),  (5)

Z-transformation transfer function of G(z) is expressed as the formula(5), and the possible amplitude response existing therein is as shown inFIG. 2 in which ω_(p) and ω_(s) represent a passband edge and a stopbandedge of the designed filter, respectively. It should be noted that theamplitude response of G(z) may be not periodic because of no anyconstraint among the interpolation factors M, P, Q. The passband andstopband edges ω_(pma) ⁽¹⁾ and ω_(sma) ⁽¹⁾ of a filter H_(ma) ⁽¹⁾(z) areas shown in FIG. 2. A distance between ω_(pma) ⁽¹⁾ and ω_(sma) ⁽¹⁾ isindicated by d₁, and a distance between ω_(pma) ⁽¹⁾ and ω_(sma) ⁽¹⁾ isindicated by d₂.

It is assumed that the passband ripple and the stopband ripple of thefilter are represented by δ_(p) and δ_(s) , respectively. Since aconventional calculation way cannot be used to determine the passbandand stopband edges of each subfilter, a new way is required to obtainthem.

The transition band of the whole filter may be formed by H_(a)⁽²⁾(z^(M)) or its complement, and therefore, there are two cases: Case Aand Case B. In the Case A, the transition band of the filter is providedby H_(a) ⁽²⁾(z^(M)); in the Case B, the transition band of the filter isprovided by the complement of H_(a) ⁽²⁾(z^(M)). Besides, masking may beachieved by H_(ma) ⁽²⁾(z^(P)) and H_(mc) ⁽²⁾(z^(Q)) or theircomplements. The following two parameters are defined to distinguish theabove cases:

when H_(ma) ⁽²⁾(z^(P)) is used, Case_(p)=A, and when H_(ma) ⁽²⁾(z^(P))is used, Case_(q)=A;

when the complement of H_(ma) ⁽²⁾(z^(P)) is used, Case_(p)=B, and whenthe complement of H_(ma) ⁽²⁾(z^(P)) is used, Case_(q)=B.

Depending on the positions of the passband and stopband edges, thereexist multiple cases for the improved structure. Now, it is first tofocus on the design in Case A since Case A and Case B are similar. A setof interpolation factors [M, P, Q] leading to the lowest complexity ofthe filter may be found out through global search, and the complexity isdecided by the number of multipliers. For the given M, P and Q,illustration will be made on how the parameters of these subfilters aredetermined below.

I. Calculation of the passband and stopband edges of a prototype filterH_(a) ⁽²⁾(z)

The passband and stopband edges of the prototype filter H_(a) ⁽²⁾(z) aredetermined by a conventional way. The H_(a) ⁽²⁾(z) passband edge θ_(a)and the stopband edge φ_(a) of H_(a) ⁽²⁾(z) are obtained readily:

For Case A,m=└ω _(p) M/2π┘,  (6a)θ_(a)=ω_(p) M−2mπ,  (6b)φ_(a)=ω_(s) M−2mπ.  (6c);

For the Case B,m=┌ω _(s) M/2π┐,  (7a)θ_(a)=2mπ−ω _(s) M,  (7b)φ_(a)=2mπ−ω _(p) M,  (7c)wherein └x┘ represents a largest integer not more than x. ┌x┐ representsa smallest integer not less than x. whether the results satisfies0<θ_(a)<φ_(a)<π is determined in the two cases respectively, and if not,the results are abandoned. The condition 0<θ_(a)<φ_(a)<π must besatisfied in both cases, and only one case meets the requirement.

II. Calculation of the passband and the stopband of second-stage maskingfilters

The passband and stopband edges of H_(ma) ⁽²⁾(z) and H_(mc) ⁽²⁾(z) arerepresented by ω_(pma) ⁽²⁾, ω_(sma) ⁽²⁾, ω_(pmc) ⁽²⁾, and ω_(smc) ⁽²⁾,respectively. In addition, passbands of H_(a) ⁽²⁾(Mω) from 0 to π in thestep (2) are orderly represented by 0, 2, . . . ,

${2 \times \left\lfloor \frac{M}{2} \right\rfloor},$and passbands of 1−H_(a) ⁽²⁾(Mω) from 0 to π are orderly represented by1, 3, . . . ,

$2 \times {\left\lfloor \frac{M - 1}{2} \right\rfloor.}$

1) Case A

The amplitude response of the up-branch of G(z) is as shown in FIG. 3.Assuming that the passband labeled as 2m of H_(a) ⁽²⁾(Mω) provides thetransition band, the passband of the masking filter from which thetransition band is extracted is defined as an “effective passband”. Withreference to FIG. 3, a restrictive condition is established to preventd₁ from being too small: the passband 2m should be at least completelyextracted. In order to prevent d₂ from being too small: a restrictivecondition is also established: the passband 2(m+1) should completelyfall outside an effective passband range. The two constraint conditionsmay also be described as follows:

1) the left passband edge x₁ of the effective passband should not begreater than the left edge ω₁ of the passband 2m;

2) the right passband edge x₂ of the effective passband should not belower than the right edge ω₂ of the passband 2m;ω₂

3) the right stopband edge x₃ of the effective passband should not begreater than the left edge ω₃ of the passband 2(m+1).

Relevant inequations and variable values are as shown in Table I inwhich

$0 \leq p \leq {\left\lfloor \frac{P}{2} \right\rfloor.}$For the known set of interpolation factors [M, P, Q], if there exists pallowing ω_(pma) ⁽²⁾ and ω_(sma) ⁽²⁾ to satisfy the three in equationsfrom (10a) to (10c), other filter parameters will be calculatedcontinuously; or otherwise, this set of interpolation factors isabandoned. After the inequations are solved, upper and lower bounds ofω_(pma) ⁽²⁾ and ω_(sma) ⁽²⁾ may be obtained. The maximum of ω_(sma) ⁽²⁾and the minimum of ω_(pma) ⁽²⁾ are taken, respectively, and then thetransition band of the filter H_(ma) ⁽²⁾(z) is the widest. The values ofω_(pma) ⁽²⁾ and ω_(sma) ⁽²⁾ are as follows:

For Case_(p)=A,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {\omega_{1}P}},{{\omega_{2}P} - {2\pi\; p}}} \right)}},} \\{\omega_{sma}^{(2)} = {{\omega_{3}P} - {2\pi\; p}}}\end{matrix}.} \right. & (8)\end{matrix}$

For Case_(p)=B,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pma}^{(2)} = {{2\pi\; p} - {\omega_{3}P}}},} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{1}P} - {2{\pi\left( {p - 1} \right)}}},{{2\pi\; p} - {\omega_{2}P}}} \right)}}\end{matrix}.} \right. & (9)\end{matrix}$

TABLE I Variable Values And Inequations For Calculating The Passband AndThe Stopband Of H_(ma) ⁽²⁾ (z) In Case A $\begin{matrix}{Inequations} & \; \\{{x_{1} \leq \omega_{1}} = \frac{{2\;\pi\; m} - \varphi_{a}}{M}} & \left( {10\; a} \right) \\{{x_{2} \geq \omega_{2}} = \omega_{p}} & \left( {10\; b} \right) \\{{x_{3} \leq \omega_{2}} = \frac{{2\;{\pi\left( {m + 1} \right)}} - \varphi_{a}}{M}} & \left( {10\; c} \right)\end{matrix}$ $\begin{matrix}{{Case}_{p} = A} & \; \\{x_{1} = {\left( {{2\;\pi\; p} - \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {11\; a} \right) \\{x_{2} = {\left( {{2\;\pi\; p} + \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {11\; b} \right) \\{x_{3} = {\left( {{2\;\pi\; p} + \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {11\; c} \right) \\{{Case}_{p} = B} & \; \\{x_{1} = {\left( {{2\;\pi\;\left( {p - 1} \right)} + \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {12\; a} \right) \\{x_{2} = {\left( {{2\;\pi\; p} - \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {12\; b} \right) \\{x_{3} = {\left( {{2\;\pi\; p} - \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {12\; c} \right)\end{matrix}$

The passband and stopband edges of H_(mc) ⁽²⁾(z) may be obtainedsimilarly. The amplitude response of the down-branch of G(z) is as shownin FIG. 4. To prevent d₁ from being too small, the passband 2(m−1) isfully reserved. To prevent d from being too small, the passband (2m+1)is fully removed. Corresponding inequations and variable values areshown in Table II in which

$0 \leq q \leq {\left\lfloor \frac{q}{2} \right\rfloor.}$If there exists the parameter q allowing ω_(pmc) ⁽²⁾ and ω_(smc) ⁽²⁾ tosatisfy the four in equations from (14a) to (14d), other filterparameters will be calculated continuously; or otherwise, this set ofinterpolation factors is abandoned. To make the transition band ofH_(mc) ⁽²⁾(z) widest, the maximum of ω_(smc) ⁽²⁾ and the minimum ofω_(pmc) ⁽²⁾ are taken, respectively.

For Case_(q)=A,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{4}Q}},{{\omega_{5}Q} - {2\pi\; q}}} \right)}},} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{6}Q} - {2\pi\; q}},{{2{\pi\left( {q + 1} \right)}} - {\omega_{7}Q}}} \right)}}\end{matrix}.} \right. & (12)\end{matrix}$

For Case_(q)=B,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{6}Q}},{{\omega_{7}Q} - {2\pi\; q}}} \right)}},} \\{\omega_{smc}^{(2)} = {\min\left( {{\omega_{4} - {2{\pi\left( {q - 1} \right)}}},{{2\pi\; q} - {\omega_{5}Q}}} \right)}}\end{matrix}.} \right. & (13)\end{matrix}$

TABLE II Variable Values And Inequations For Calculating The PassbandAnd The Stopband Of H_(ma) ⁽²⁾ (z) In Case A $\begin{matrix}{Inequations} & \; \\{{y_{1} \leq \omega_{4}} = \frac{{2\;{\pi\left( {m - 1} \right)}} + \varphi_{a}}{M}} & \left( {14\; a} \right) \\{{y_{2} \geq \omega_{5}} = \frac{{2\;\pi\; m} - \theta_{a}}{M}} & \left( {14\; b} \right) \\{{y_{3} \leq \omega_{6}} = \omega_{s}} & \left( {14\; c} \right) \\{{y_{4} \geq \omega_{7}} = \frac{{2\;{\pi\left( {m + 1} \right)}} - \theta_{a}}{M}} & \left( {14\; d} \right)\end{matrix}$ $\begin{matrix}{{Case}_{q} = A} & \; \\{y_{1} = {\left( {{2\;\pi\; q} - \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {15\; a} \right) \\{y_{2} = {\left( {{2\;\pi\; q} + \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {15\; b} \right) \\{y_{3} = {\left( {{2\;\pi\; q} + \omega_{ama}^{(2)}} \right)\text{/}Q}} & \left( {15\; c} \right) \\{y_{4} = {\left( {{2\;{\pi\left( {q + 1} \right)}} - \omega_{ama}^{(2)}} \right)\text{/}Q}} & \left( {15\; d} \right)\end{matrix}$ $\begin{matrix}{{Case}_{q} = B} & \; \\{y_{1} = {\left( {{2\;{\pi\left( {q - 1} \right)}} + \omega_{sma}^{(2)}} \right)\text{/}Q}} & \left( {16\; a} \right) \\{y_{2} = {\left( {{2\;\pi\; q} - \omega_{ama}^{(2)}} \right)\text{/}Q}} & \left( {16\; b} \right) \\{y_{3} = {\left( {{2\;\pi\; q} - \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {16\; c} \right) \\{y_{4} = {\left( {{2\;\pi\; q} + \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {16\; d} \right)\end{matrix}$

2) Case B

For Case B, the transition band of G(z) is formed by the complement ofH_(a) ⁽²⁾(z^(M)).

Determination of the passband and stopband edges of H_(ma) ⁽²⁾(z) issimilar to solving of the passband and stopband edges of H_(mc) ⁽²⁾(z)in Case A. Corresponding inequations and parameters are as shown inTable III. The passband and stopband edges of H_(ma) ⁽²⁾(z) are shownbelow:

For Case_(p)=A,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {P\;\omega_{4}}},{{\omega_{5}P} - {2\pi\; p}}} \right)}},} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{6}P} - {2\pi\; p}},{{2{\pi\left( {p + 1} \right)}} - {\omega_{7}P}}} \right)}}\end{matrix}.} \right. & (17)\end{matrix}$

For Case_(p)=B,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - \;{\omega_{6}P}},{{\omega_{7}P} - {2\pi\; p}}} \right)}},} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{4}P} - {2{\pi\left( {p - 1} \right)}}},{{2\pi\; p} - {\omega_{5}P}}} \right)}}\end{matrix}.} \right. & (18)\end{matrix}$

TABLE III Variable Values And Inequations For Calculating The PassbandAnd The Stopband Of H_(ma) ⁽²⁾ (z) in Case B $\begin{matrix}{Inequations} & \; \\{{y_{1} \leq \omega_{4}} = \frac{{2\;{\pi\left( {m - 1} \right)}} - \theta_{a}}{M}} & \left( {19\; a} \right) \\{{y_{2} \geq \omega_{5}} = \frac{{2\;{\pi\left( {m - 1} \right)}} + \varphi_{a}}{M}} & \left( {19\; b} \right) \\{{y_{3} \leq \omega_{6}} = \omega_{s}} & \left( {19\; c} \right) \\{{y_{4} \geq \omega_{7}} = \frac{{2\;\pi\; m} + \varphi_{a}}{M}} & \left( {19\; d} \right)\end{matrix}$ $\begin{matrix}{{Case}_{p} = A} & \; \\{y_{1} = {\left( {{2\;\pi\; p} - \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {20\; a} \right) \\{y_{2} = {\left( {{2\;\pi\; p} + \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {20\; b} \right) \\{y_{3} = {\left( {{2\;\pi\; p} + \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {20\; c} \right) \\{y_{4} = {\left( {{2\;{\pi\left( {p + 1} \right)}} - \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {20\; d} \right)\end{matrix}$ $\begin{matrix}{{Case}_{p} = B} & \; \\{y_{1} = {\left( {{2\;{\pi\left( {p - 1} \right)}} + \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {21\; a} \right) \\{y_{2} = {\left( {{2\;\pi\; p} - \omega_{sma}^{(2)}} \right)\text{/}P}} & \left( {21\; b} \right) \\{y_{3} = {\left( {{2\;\pi\; p} - \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {21\; c} \right) \\{y_{4} = {\left( {{2\;\pi\; p} + \omega_{pma}^{(2)}} \right)\text{/}P}} & \left( {21\; d} \right)\end{matrix}$

Determination of the passband and stopband edges of H_(mc) ⁽²⁾(z) issimilar to the method of calculating the passband and stopband edges ofH_(ma) ⁽²⁾(z) in Case A. Corresponding inequations and parameters areshown in Table IV. The passband and stopband edges of H_(mc) ⁽²⁾(z) areshown below:

For Case_(q)=A,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{1}Q}},{{\omega_{2}Q} - {2\pi\; q}}} \right)}},} \\{\omega_{smc}^{(2)} = {{\omega_{3}Q} - {2\pi\; q}}}\end{matrix}.} \right. & (22)\end{matrix}$

For Case_(q)=B,

$\begin{matrix}\left\{ {\begin{matrix}{{\omega_{pmc}^{(2)} = {{2\pi\; q} - {\omega_{3}Q}}},} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{1}Q} - {2{\pi\left( {q - 1} \right)}}},{{2\pi\; q} - {\omega_{2}Q}}} \right)}}\end{matrix}.} \right. & (23)\end{matrix}$

TABLE IV Variable Values And Inequations For Calculating The PassbandAnd The Stopband Of H_(ma) ⁽²⁾ (z) In Case B $\begin{matrix}{Inequations} & \; \\{{x_{1} \leq \omega_{1}} = \frac{{2\;{\pi\left( {m - 3} \right)}} + \theta_{a}}{M}} & \left( {24\; a} \right) \\{{x_{2} \geq \omega_{2}} = \omega_{s}} & \left( {24\; b} \right) \\{{x_{3} \leq \omega_{3}} = \frac{{2\;\pi\; m} + \theta_{a}}{M}} & \left( {24\; c} \right) \\{{Case}_{q} = A} & \; \\{x_{1} = {\left( {{2\;\pi\; q} - \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {25\; a} \right) \\{x_{2} = {\left( {{2\;\pi\; q} + \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {25\; b} \right) \\{x_{3} = {\left( {{2\;\pi\; q} + \omega_{sma}^{(2)}} \right)\text{/}Q}} & \left( {25\; c} \right)\end{matrix}$ $\begin{matrix}{{Case}_{q} = B} & \; \\{x_{1} = {\left( {{2\;{\pi\left( {q - 1} \right)}} + \omega_{sma}^{(2)}} \right)\text{/}Q}} & \left( {26\; a} \right) \\{x_{2} = {\left( {{2\;\pi\; q} - \omega_{sma}^{(2)}} \right)\text{/}Q}} & \left( {26\; b} \right) \\{x_{3} = {\left( {{2\;\pi\; q} - \omega_{pma}^{(2)}} \right)\text{/}Q}} & \left( {26\; c} \right)\end{matrix}$

III. Calculation of the passband and stopband edges of first-stagemasking filters

1) Case A

For a masking filter H_(ma) ⁽¹⁾(z), since the transition band of H(z) isprovided by H_(a) ⁽²⁾(z^(M)), its passband edge ω_(pma) ⁽¹⁾ is equal toω_(p). Determination of the stopband edges ω_(sma) ⁽¹⁾, ω_(sma) ⁽¹⁾ isas shown in FIG. 5.

The stopband edge ω_(sma) ⁽¹⁾ is the right endpoint of d₂. Therefore,the first masking filter passband on the right of the transition band isfocused. The left stopband cutoff points of the two passbands aredenoted as t₁ and t₂. It is required to find out the position of t₁ inH_(a) ⁽²⁾(Mω) and the position of t₂ in 1−H_(a) ⁽²⁾(Mω). The passbandand stopband regions of H_(a) ⁽²⁾(Mω) may be obtained through thefollowing formulas:

$\begin{matrix}{{{R_{pass}(k)} = \left\lbrack {\frac{{2\pi\; k} - \theta_{a}}{M},\frac{{2\pi\; k} + \theta_{a}}{M}} \right\rbrack},{k = 0},\ldots\mspace{14mu},\left\lfloor {M/2} \right\rfloor,} & (27) \\{{{R_{stop}(k)} = \left\lbrack {\frac{{2\pi\;\left( {k - 1} \right)} + \varphi_{a}}{M},\frac{{2\pi\; k} + \varphi_{a}}{M}} \right\rbrack},{k = 0},\ldots\mspace{14mu},\left\lfloor {M/2} \right\rfloor,} & (28)\end{matrix}$

A temporary value of ω_(sma) ⁽¹⁾ is obtained according to the positionof t₁, and denoted as ω_(sma) _(_) _(temp1) ⁽¹⁾:

$\begin{matrix}{\omega_{{sma}\;\_\;{temp}\; 1}^{(1)} = \left\{ {\begin{matrix}t_{1} & {t_{1} \notin {R_{stop}(k)}} \\\omega_{8} & {t_{1} \in {R_{stop}\left( k_{1} \right)}}\end{matrix},} \right.} & (29)\end{matrix}$wherein ω₈ is the right endpoint of R_(stop)(k₁), and k₁ is an integersatisfying t₁ϵR_(stop)(k₁).

Another temporary value of ω_(sma) ⁽¹⁾ is obtained according to theposition of t₂, and denoted as ω_(sma) _(_) _(temp2) ⁽¹⁾:

$\begin{matrix}{\omega_{{sma}\;\_\;{temp}\; 2}^{(1)} = \left\{ {\begin{matrix}t_{2} & {t_{2} \notin {R_{pass}(k)}} \\\omega_{9} & {t_{2} \in {R_{pass}\left( k_{2} \right)}}\end{matrix},} \right.} & (30)\end{matrix}$wherein ω₉ is the right endpoint of R_(pass)(k₂) and k₂ is an integersatisfying t₂ϵR_(stop)(k₂).

The value of ω_(sma) ⁽¹⁾ is obtained through the following formula:ω_(sma) ⁽¹⁾=min(ω_(sma) _(_) _(temp1) ⁽¹⁾, ω_(sma) _(_) _(temp2)⁽¹⁾),  (31)wherein parameters t₁, t₂, ω₈ and ω₉ are as shown in Table V.

TABLE V Variable Values For Calculating ω_(sma) ⁽¹⁾ in Case A Case_(p) =A t₁ = (2π(p + 1) − ω_(sma) ⁽²⁾)/P (32) Case_(p) = B t₁ = (2πp + ω_(pma)⁽²⁾)/P (33) Case_(q) = A t₂ = (2π(q + 1) − ω_(smc) ⁽²⁾)/Q (34) Case_(q)= B t₂ = (2πq + ω_(pmc) ⁽²⁾)/Q (35) ω₈ = (2πk₁ − φ_(a))/M (36) ω₉ =(2πk₂ + θ_(a))/M (37)

For a masking filter H_(mc) ⁽¹⁾(z), its stopband edge ω_(smc) ⁽¹⁾ isequal to ω_(s), and only its passband edge ω_(pmc) ⁽¹⁾ needs to bedetermined. Calculation of ω_(pmc) ⁽¹⁾ is as shown in FIG. 6. Sinceω_(pmc) ⁽¹⁾ is the left endpoint of d₁, for H_(ma) ⁽²⁾(Pω), only thepassband including the transition band is focused, and for H_(mc)⁽²⁾(Qω), only the first passband on the left of the transition band isfocused. The left passband edges of the two passbands are defined as t₃and t₄.

If t₃≥t₄, then

$\begin{matrix}{\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{3} & {t_{3} \notin {R_{stop}(k)}} \\{\max\left( {\frac{{2\pi\;\left( {k_{3} - 1} \right)} + \varphi_{a}}{M},t_{4}} \right)} & {t_{3} \in {R_{stop}\left( k_{3} \right)}}\end{matrix};} \right.} & (38)\end{matrix}$

if t₃<t₄, then

$\begin{matrix}{\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{4} & {t_{4} \notin {R_{pass}(k)}} \\{\max\left( {\frac{{2\pi\; k_{4}} - \theta_{a}}{M},t_{3}} \right)} & {t_{4} \in {R_{pass}\left( k_{4} \right)}}\end{matrix},} \right.} & (39)\end{matrix}$wherein the values of the parameters t₃ and t₄ are as shown in Table VI.

TABLE V Variable Values For Calculating ω_(pmc) ⁽¹⁾ in Case A Case_(p) =A t₂ = (2πp − ω_(pma) ⁽²⁾)/P (40) Case_(p) = B t₂ = (2π(p − 1) + ω_(sma)⁽²⁾)/P (41) Case_(q) = A t₄ = (2πq − ω_(pmc) ⁽²⁾)/Q (42) Case_(q) = B t₄= (2π(q − 1) + ω_(smc) ⁽²⁾)/Q (43)

2) Case B

In Case B, the ways of calculating ω_(sma) ⁽¹⁾ and ω_(pmc) ⁽¹⁾ aresimilar to those in Case A with corresponding parameters as shown inTable VII and Table VIII. It should be noted that the passband andstopband edges of all the subfilters must fall into the range from 0 toπ; otherwise, the subsequent design of the set of interpolation factors[M, P, Q] is considered to be meaningless.

TABLE VII Variable Values For Calculating ω_(sma) ⁽¹⁾ in Case B Case_(p)= A t₂ = (2π(p + 1) − ω_(sma) ⁽²⁾)/P  44) Case_(p) = B t₂ = (2πp +ω_(pma) ⁽²⁾)/P (45) Case_(q) = A t₁ = (2π(q + 1) − ω_(sma) ⁽²⁾)/Q (46)Case_(q) = B t₁ = (2πq + ω_(pmc) ⁽²⁾)/Q (47) ω₈ = (2πk₁ + θ_(a))/M (48)ω₉ = (2πk₂ − φ_(a))/M (49)

TABLE VIII Variable Values For Calculating ω_(pmc) ⁽¹⁾ in Case BCase_(p) = A t₄ = (2πp − ω_(pma) ⁽²⁾)/P (50) Case_(p) = B t₄ = (2π(p− 1) + ω_(sma) ⁽²⁾)/P (51) Case_(q) = A t₂ = (2πq − ω_(pmc) ⁽²⁾)/Q (52)Case_(q) = B t₂ = (2π(q − 1) + ω_(smc) ⁽²⁾)/Q (53)

Optimization Method

Step 1

Some sets having fewer multipliers are selected from all the effectivesets of interpolation factors as optimization objects (generally notmore than 105% of the lowest number of multipliers. Calculation of thenumber of multipliers is carried out through the following formula:{circumflex over (N)} _(mult)=({circumflex over (N)} _(a) +{circumflexover (N)} _(ma2) +{circumflex over (N)} _(mc2))/2+└({circumflex over(N)} _(ma)+2)/2┘+└({circumflex over (N)} _(mc)+2)/2┘+3,  (54)wherein {circumflex over (N)}_(a), {circumflex over (N)}_(ma2),{circumflex over (N)}_(mc2), {circumflex over (N)}_(ma) and {circumflexover (N)}_(mc) are filter orders estimated for H_(a) ⁽²⁾(z), H_(ma)⁽²⁾(z), H_(mc) ⁽²⁾(z), H_(ma) ⁽¹⁾(z), and H_(mc) ⁽¹⁾(z), respectively,and wherein the orders may be obtained) by using the firpmord functionin matlab.

Step 2

Each subfilter is designed according to the Parks-McClellan algorithm.As a matter of experience, after optimization, the order N_(a) of H_(a)⁽²⁾(z) is equal to the estimated order {circumflex over (N)}_(a), whilethe orders N_(ma2), N_(mc2), N_(ma) and N_(mc) of the masking filtersare 60% of the respective estimated orders. When the order of asubfilter is not an integer, it is rounded. It should be noted that theorder of the prototype filter and the orders of the second-stage maskingfilters must be even numbers, while the orders of the two first-stagemasking filters are simultaneously odd numbers or even numbers.

Step 3

The subfilters are simultaneously optimized by using a nonlinearoptimization algorithm. A vector Ø defined as a combination ofparameters of the subfilters is as shown below:

$\begin{matrix}{\varnothing = {\left\lbrack {{h_{ma}^{(1)}(0)},\ldots\mspace{14mu},{h_{ma}^{(1)}\left( \left\lfloor \frac{N_{ma}}{2} \right\rfloor \right)},{h_{m\; c}^{(1)}(0)},\ldots\mspace{14mu},{h_{m\; c}^{(1)}\left( \left\lfloor \frac{N_{m\; c}}{2} \right\rfloor \right)},{h_{ma}^{(2)}(0)},\ldots\mspace{14mu},{h_{ma}^{(2)}\left( \frac{N_{{ma}\; 2}}{2} \right)},{h_{m\; c}^{(2)}(0)},\ldots\mspace{14mu},{h_{m\; c}^{(2)}\left( \frac{N_{m\; c\; 2}}{2} \right)},{h_{a}^{(2)}(0)},\ldots\mspace{14mu},{h_{a}^{(2)}\left( \frac{N_{a}}{2} \right)}} \right\rbrack.}} & (55)\end{matrix}$

To optimize the filter parameters, the following object function shouldbe minimized:

$\begin{matrix}{{E = {\max\limits_{\omega \in {{\lbrack{0,\omega_{p}}\rbrack}\bigcup{\lbrack{\omega_{s},\pi}\rbrack}}}\;{{{W(\omega)}\left\lbrack {{H\left( {\varnothing,\omega} \right)} - {D(\omega)}} \right\rbrack}}}},} & (56)\end{matrix}$wherein H(Ø, ω) is the zero phase frequency response of the systemfunction H(z);

D(ω) is an ideal zero phase frequency function; W(ω) is a weightedvector. For the passbands ωϵ[0, ω_(p)], W(ω) and D(ω) are equal to 1;for the stopbands ωϵ[ω_(s), π], W(ω) is equal to δ_(p)/δ_(s), and D(ω)is equal to 0.

This is a problem of minimization and maximization. The fminimaxfunction in the optimization toolbox provided by MathWorks Company cansolve this problem. To achieve more efficient optimization, non-uniformfrequency points are employed; the closer to the regions of the passbandedge and the stopband edge, the denser the frequency points. It issuggested that the frequency points within 10% region closest to thepassband edge account for 25% of the total number of points, and it isthe same as the stopband edge. During optimization using the fminimaxfunction, the parameter requirements of a filter can be met only when Eis not greater than δ_(p).

In order to find out the optimal solutions, for each set ofinterpolation factors, the order of each subfilter needs to be variednear the respective estimated order before optimization. As a matter ofexperience, the order of H_(a) ⁽²⁾(z) is not varied, and variation isonly required to be made to the order of each masking filter. ForN_(ma2), N_(mc2), N_(ma) and N_(mc), a range of variation is 4; thevariation of N_(ma2) and N_(mc2) is 2, while the variation of N_(ma) andN_(mc) is 1. After optimization, there may exist multiple solutionssatisfying filter specifications, and the one having the fewestmultipliers is selected as the optimal solution. In case of multipleoptimal solutions, the one having the smallest time delay is selected.

Example 1

The specific specifications of a filter having an extremely narrowtransition band are as follows: ω_(p)=0.6π, ω_(s)=0.602π, δ_(p)=0.01,δ_(s)=0.01. By using the present design method, the final results are asfollows: N_(a)=28, N_(ma2)=20, N_(mc2)=16, N_(ma)=17, N_(mc)=29. Thethree interpolation factors M, P, Q are 69,9,9, respectively, which fallinto Case B type: Case_(p)=B, and Case_(q)=B. The number of multipliersis 59 with a group time delay being 1070.5. The passband and stopbandripples are 0.01 and 0.00999, respectively. Comparison of results of themethod of the present invention with those of other existing methods isas shown in Table IX.

TABLE IX Results of Designing Filter Using Various Methods in Example 1And Comparison Thereof Total Group Number Of Time Method MultipliersDelay Conventional 92 1105 two-stage FRM SFFM-FRM case A 86 1638.5SFFM-FRM case B 84 2150.5 Serial-masking FRM 83 1016 Non-periodical FRM55 1214 Improved method 59 1070.5

As can be seen from Table IX, when compared with the conventionaltwo-stage FRM, SFFM and serial-masking FRM methods, the method providedby the present invention is lower in complexity. When compared with theconventional two-stage FRM, the complexity is reduced by 35.8%; whencompared with the SFFM-FRM, it is reduced by 31.4%; when compared with aserial-masking filter, it is reduced by 28.9%. When compared with theNon-periodical FRM, although the number of multipliers is increased by7.3%, the group time delay is reduced by 11.8%. The time delay in theresult of the present method is the smallest when compared with othermethods apart from the Serial-masking FRM. Although the time delay isincreased by 5.1% in contrast with the Serial-masking FRM, the number ofmultipliers is reduced by 28.9%.

The constraint conditions among the interpolation factors of aconventional FRM filter are broken through a new design method, and thesubfilters are optimized simultaneously by means of a nonlinear jointoptimization method. Results indicate that, as compared to aconventional design method of a two-stage FRM filter, a filter designedusing the improved method is lower in complexity.

While the specific embodiments of the present invention are describedabove in conjunction with the accompanying drawings, they are notlimitations to the protection scope of the present invention. It shouldbe understood by those skilled in the relevant art that variousmodifications or variations made by those skilled in the art withoutcreative work on the basis of the technical scheme of the presentinvention still fall into the protection range of the present invention.

The invention claimed is:
 1. A method of constructing a two-stage FRMfilter, comprising the following steps of: determining, with aprocessor, a transfer function H(z) of a two-stage FRM filter asfollows:H(z)=G(z)H _(ma) ⁽¹⁾(z)+(1−G(z))H _(mc) ⁽¹⁾(z), wherein G(z)=H_(a)⁽²⁾(z^(M))H_(ma) ⁽²⁾(z^(P))+(1−H_(a) ⁽²⁾(z^(M)))H_(mc) ⁽²⁾(z^(Q)),without any constraint among interpolation factors M, P, Q; H_(a)⁽²⁾(z^(M)) represents a prototype filter, while H_(ma) ⁽¹⁾(z) and H_(mc)⁽¹⁾(z) represent first-stage masking filters, respectively, and H_(ma)⁽²⁾(z^(P)) and H_(mc) ⁽²⁾(z^(Q)) represent second-stage masking filters,respectively; storing the interpolation factors M, P, Q in a memory;searching, with a processor, within a search range for [M, P, Q], andfor a certain set [M, P, Q], calculating passband and stopband edgeparameters of the prototype filter H_(a) ⁽²⁾(z), passband and stopbandedge parameters of the second-stage masking filters and passband andstopband edge parameters of the first-stage masking filters in Case Aand Case B, respectively, on the basis that a transition band of thewhole filter is provided by H_(a) ⁽²⁾(z^(M)) or a complement of H_(a)⁽²⁾(z^(M)); calculating calculating, with a processor, the complexity ofthe FRM filter according to the calculated passband and stopband edgeparameters, and finding out one or more sets [M, P, Q] having the lowestcomplexity within the search range; optimizing, with a processor, animproved FRM filter according to the calculated filter parameters, bydetermining a number of multipliers implementing the improved FRM filterand a group time delay performed by the improved FRM filter, andconstructing the improved FRM filter by arranging the multipliers, basedon the calculated filter parameters.
 2. The method of constructing atwo-stage FRM filter according to claim 1, wherein a way of calculatingthe passband edge θ_(a) and the stopband edge φ_(a) of the prototypefilter H_(a) ⁽²⁾(z^(M)) in the step (2) is as follows: when thetransition band of the whole filter is H_(a) ⁽²⁾(z^(M)), i.e., in theCase A:m=└ω _(p) M/2π┘,θ_(a)=ω_(p) M−2mπ,φ_(a)=ω_(s) M−2mπ; when the transition band of the whole filter is thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B:m=┌ω _(s) M/2π┐,θ_(a)=2mπ−ω _(s) M,φ_(a)=2mπ−ω _(p) M; wherein └x┘ represents a largest integer not morethan x; ┌x┐ represents a smallest integer not less than x; whether theresult satisfies 0<θ_(a)<φ_(a)<π is determined in the two casesrespectively, and if not, the results are abandoned.
 3. The method ofconstructing a two-stage FRM filter according to claim 1, whereinpassbands of H_(a) ⁽²⁾(Mω) from 0 to π in the step (2) are orderlyrepresented by 0, 2, . . . ,${2 \times \left\lfloor \frac{M}{2} \right\rfloor},$ and passbands of1−H_(a) ⁽²⁾(Mω) from 0 to π are orderly represented by 1, 3, . . . ,${2 \times \left\lfloor \frac{M - 1}{2} \right\rfloor};$ assuming thatthe passband labeled as 2m of H_(a) ⁽²⁾(Mω) provides the transitionband, the passband of the masking filter from which the transition bandis extracted is defined as an effective passband; and in order to reducethe complexity of the two first-stage masking filters, the followingrestrictive conditions are established: the passband 2m should be atleast completely extracted; the passband 2(m+1) should be completelyfall outside an effective passband range.
 4. The method of constructinga two-stage FRM filter according to claim 1, wherein when the transitionband of the whole filter is provided by H_(a) ⁽²⁾(z^(M)), i.e., in theCase A, a way of calculating the passband edge ω_(pma) ⁽²⁾ and thestopband edge ω_(sma) ⁽²⁾ of the second-stage masking filter H_(ma)⁽²⁾(z), is as follows: (1) when masking is provided by H_(ma)⁽²⁾(z^(P)), the case is denoted as Case_(p)=A: $\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - \;{\omega_{1}P}},{{\omega_{2}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {{\omega_{3}P} - {2\pi\; p}}}\end{matrix};} \right.$ (2) when masking is provided by the complementof H_(ma) ⁽²⁾(z^(P)), the case is denoted as Case_(p)=B:$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {{2\pi\; p} - {\omega_{3}P}}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{1}P} - {2{\pi\left( {p - 1} \right)}}},{{2\pi\; p} - {\omega_{2}P}}} \right)}}\end{matrix};} \right.$ wherein ω₁ is a left edge of the passband 2m ofthe prototype filter, while ω₂ is a right edge of the passband 2m of theprototype filter, and ω₃ is a left edge of the passband 2(m+1) of theprototype filter; P is the interpolation factor, which is a given value;p is an integer, and should satisfy the following condition:${0 \leq p \leq \left\lfloor \frac{P}{2} \right\rfloor};$ if no psatisfying the condition exists, the set of parameters is abandoned. 5.The method of constructing a two-stage FRM filter according to claim 1,wherein when the transition band of the whole filter is provided byH_(a) ⁽²⁾(z^(M)), i.e., in the Case A, a way of calculating the passbandedge ω_(pmc) ⁽²⁾ and the stopband edge ω_(smc) ⁽²⁾ of the second-stagemasking filter H_(mc) ⁽²⁾(z) is as follows: (1) when masking is providedby H_(mc) ⁽²⁾(z^(Q)), the case is denoted as Case_(q)=A:$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{4}Q}},{{\omega_{5}Q} - {2\pi\; q}}} \right)}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{6}Q} - {2\pi\; q}},{{2{\pi\left( {q + 1} \right)}} - {\omega_{7}Q}}} \right)}}\end{matrix};} \right.$ (2) when masking is provided by the complementof H_(mc) ⁽²⁾(z^(Q)), the case is denoted as Case_(q)=B:$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{6}Q}},{{\omega_{7}Q} - {2\pi\; q}}} \right)}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{4}Q} - {2{\pi\left( {q - 1} \right)}}},{{2\pi\; q} - {\omega_{5}Q}}} \right)}}\end{matrix};} \right.$ wherein ω₄ is a left passband edge of thecomplementary filter passband 2m−1 of the prototype filter; ω₅ is aright stopband edge of the complementary filter passband 2m−1 of theprototype filter; ω₆ is a left passband edge of the complementary filterpassband 2m+1 of the prototype filter; ω₇ is a right stopband edge ofthe complementary filter passband 2m+1 of the prototype filter; q is aninteger, and should satisfy the following condition:${0 \leq q \leq \left\lfloor \frac{q}{2} \right\rfloor};$ if no qsatisfying the condition exists, the set of parameters is abandoned. 6.The method of constructing a two-stage FRM filter according to claim 1,wherein when the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B, a way ofcalculating the passband edge ω_(pma) ⁽²⁾ and the stopband edge ω_(sma)⁽²⁾ of the second-stage masking filter H_(ma) ⁽²⁾(z) is as follows: (1)when masking is provided by H_(ma) ⁽²⁾(z^(P)), the case is denoted asCase_(p)=A: $\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {P\;\omega_{4}}},{{\omega_{5}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{6}P} - {2\pi\; p}},{{2{\pi\left( {p + 1} \right)}} - {\omega_{7}P}}} \right)}}\end{matrix};} \right.$ (2) when masking is provided by the complementof H_(ma) ⁽²⁾(z^(P)), the case is denoted as Case_(p)=B:$\left\{ {\begin{matrix}{\omega_{pma}^{(2)} = {\max\left( {{{2\pi\; p} - {\omega_{6}P}},{{\omega_{7}P} - {2\pi\; p}}} \right)}} \\{\omega_{sma}^{(2)} = {\min\left( {{{\omega_{4}P} - {2{\pi\left( {p - 1} \right)}}},{{2\pi\; p} - {\omega_{5}P}}} \right)}}\end{matrix};} \right.$ wherein ω₄ is a left passband edge of thepassband 2(m−1) of the prototype filter; ω₅ is a right stopband edge ofthe passband 2(m−1) of the prototype filter; ω₆ is a left passband edgeof the passband 2m of the prototype filter; ω₇ is a right stopband edgeof the passband 2m of the prototype filter; p is an integer, and shouldsatisfy the following condition:${0 \leq p \leq \left\lfloor \frac{P}{2} \right\rfloor};$ if no psatisfying the condition exists, the set of parameters is abandoned. 7.The method of constructing a two-stage FRM filter according to claim 1,wherein when the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., the Case B, a way of calculatingthe passband edge ω_(pmc) ⁽²⁾ and the stopband edge ω_(smc) ⁽²⁾ of thesecond-stage masking filter H_(mc) ⁽²⁾(z) is as follows: (1) whenmasking is provided by H_(mc) ⁽²⁾(z^(Q)), the case is denoted asCase_(q)=A: $\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {\max\left( {{{2\pi\; q} - {\omega_{1}Q}},{{\omega_{2}Q} - {2\pi\; q}}} \right)}} \\{\omega_{smc}^{(2)} = {{\omega_{3}Q} - {2\pi\; q}}}\end{matrix};} \right.$ (2) when masking is provided by the complementof H_(mc) ⁽²⁾(z^(Q)), the case is denoted as Case_(q)=B:$\left\{ {\begin{matrix}{\omega_{pmc}^{(2)} = {{2\pi\; q} - {\omega_{3}Q}}} \\{\omega_{smc}^{(2)} = {\min\left( {{{\omega_{1}Q} - {2{\pi\left( {q - 1} \right)}}},{{2\pi\; q} - {\omega_{2}Q}}} \right)}}\end{matrix};} \right.$ wherein ω₁ is a left stopband edge of thecomplementary filter passband 2m−1 of the prototype filter; ω₂ is aright passband edge of the complementary filter passband 2m−1 of theprototype filter; ω₃ is a left stopband edge of the complementary filterpassband 2m+1 of the prototype filter; q is an integer, and shouldsatisfy the following condition:${0 \leq q \leq \left\lfloor \frac{q}{2} \right\rfloor};$ if no qsatisfying the condition exists, the set of parameters is abandoned. 8.The method of constructing a two-stage FRM filter according to claim 1,wherein when the transition band of the whole filter is provided byH_(a) ⁽²⁾(z^(M)), i.e., in the Case A: the passband edge ω_(pma) ⁽¹⁾ ofthe first-stage masking filter H_(ma) ⁽¹⁾(z) is equal to ω_(p); a way ofcalculating the stopband edge ω_(sma) ⁽¹⁾ of the first-stage maskingfilter H_(ma) ⁽¹⁾(z) is as follows:ω_(sma)⁽¹⁾ = min (ω_(sma _ temp 1)⁽¹⁾, ω_(sma _ temp 2)⁽¹⁾), wherein$\omega_{{sma}\;\_\;{temp}\; 1}^{(1)} = \left\{ {\begin{matrix}t_{1} & {t_{1} \notin {R_{stop}(k)}} \\\omega_{8} & {t_{1} \in {R_{stop}\left( k_{1} \right)}}\end{matrix},{\omega_{{sma}\;\_\;{temp}\; 2}^{(1)} = \left\{ {\begin{matrix}t_{2} & {t_{2} \notin {R_{pass}(k)}} \\\omega_{9} & {t_{2} \in {R_{pass}\left( k_{2} \right)}}\end{matrix},{{\omega_{8} = {\left( {{2\pi\; k_{1}} = \varphi_{a}} \right)/M}};{\omega_{9} = {\left( {{2\pi\; k_{2}} - \theta_{a}} \right)/M}};{{R_{pass}(k)} = \left\lbrack {\frac{{2\pi\; k} - \theta_{a}}{M},\frac{{2\pi\; k} + \theta_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;{{R_{stop}(k)} = \left\lbrack {\frac{{2\pi\;\left( {k - 1} \right)} + \varphi_{a}}{M},\frac{{2\pi\; k} + \varphi_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;}} \right.}} \right.$(1) when H_(ma) ⁽²⁾(z^(P)) is used for masking unnecessary band of anup-branch,t ₁=(2π(p+1)−ω_(sma) ⁽²⁾)/P; (2) when the complement of H_(ma)⁽²⁾(z^(P)) is used for masking the unnecessary band of the up-branch,t ₁=(2πp+ω _(pma) ⁽²⁾)/P; (3) when H_(mc) ⁽²⁾(Z^(Q)) is used for maskingunnecessary band of a down-branch,t ₂=(2π(q+1)−ω_(smc) ⁽²⁾)/Q; (4) when a down-branch complement H_(mc)⁽²⁾(Z^(Q)) is used for masking unnecessary band,t ₂=(2πq+ω _(pmc) ⁽²⁾)/Q; wherein P, Q, M are interpolation factors; ω₈is a right endpoint of R_(stop)(k₁); k₁ is an integer satisfyingt₁∈R_(stop)(k₁); ω₉ is a right endpoint of R_(pass)(k₂); k₂ is aninteger satisfying t₂∈R_(stop)(k₂); θ_(a) is a passband edge of H_(a)⁽²⁾(z); H_(ma) ⁽¹⁾(z)φ_(a) is a stopband edge of H_(a) ⁽²⁾(z).
 9. Themethod of constructing a two-stage FRM filter according to claim 1,wherein when the transition band of the whole filter is provided by thecomplement of H_(a) ⁽²⁾(z^(M)), i.e., in the Case B: the passband edgeω_(pma) ⁽¹⁾ of the first-stage masking filter H_(ma) ⁽¹⁾(z) is equal toω_(p); a way of calculating the stopband edge ω_(sma) ⁽¹⁾ of thefirst-stage masking filter H_(ma) ⁽¹⁾(z) is as follows:ω_(sma)⁽¹⁾ = min (ω_(sma _ temp 1)⁽¹⁾, ω_(sma _ temp 2)⁽¹⁾), wherein$\omega_{{sma}\;\_\;{temp}\; 1}^{(1)} = \left\{ {\begin{matrix}t_{1} & {t_{1} \notin {R_{stop}(k)}} \\\omega_{8} & {t_{1} \in {R_{stop}\left( k_{1} \right)}}\end{matrix},{\omega_{{sma}\;\_\;{temp}\; 2}^{(1)} = \left\{ {\begin{matrix}t_{2} & {t_{2} \notin {R_{pass}(k)}} \\\omega_{9} & {t_{2} \in {R_{pass}\left( k_{2} \right)}}\end{matrix},{{\omega_{8} = {\left( {{2\pi\; k_{1}} = \theta_{a}} \right)/M}};{\omega_{9} = {\left( {{2\pi\; k_{2}} - \varphi_{a}} \right)/M}};{{R_{pass}(k)} = \left\lbrack {\frac{{2\pi\; k} - \theta_{a}}{M},\frac{{2\pi\; k} + \theta_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;{{R_{stop}(k)} = \left\lbrack {\frac{{2\pi\;\left( {k - 1} \right)} + \varphi_{a}}{M},\frac{{2\pi\; k} + \varphi_{a}}{M}} \right\rbrack}},{k = 0},\ldots\mspace{14mu},{\left\lfloor {M/2} \right\rfloor;}} \right.}} \right.$(1) when H_(mc) ⁽²⁾(z^(Q)) is used for masking unnecessary band,t ₁=(2π(q+1)−ω_(smc) ⁽²⁾)/Q; (2) when the complement of H_(mc)⁽²⁾(z^(Q)) is used for masking unnecessary band,t ₁=(2πq+ω _(pmc) ⁽²⁾)/Q; (3) when H_(ma) ⁽²⁾(z^(P)) is used for maskingunnecessary band,t ₂=(2π(p+1)−ω_(sma) ⁽²⁾)/P; (4) when the complement of H_(ma)⁽²⁾(z^(P)) is used for masking unnecessary band,t ₂=(2πp+ω _(pma) ⁽²⁾)/P; wherein ω₈ is a right endpoint ofR_(stop)(k₁); k₁ is an integer satisfying t₁∈R_(stop)(k₁); ω₉ is a rightendpoint of R_(pass)(k₂); k₂ is an integer satisfying t₂∈R_(stop)(k₂);P, Q, M are interpolation factors; θ_(a) is a passband edge of H_(a)⁽²⁾(z); H_(ma) ⁽¹⁾(z)φ_(a) is a stopband edge of H_(a) ⁽²⁾(z).
 10. Themethod of constructing a two-stage FRM filter according to claim 1,wherein (1) when the transition band of the whole filter is provided byH_(a) ⁽²⁾(z^(M)), i.e., in the Case A: the stopband edge ω_(smc) ⁽¹⁾ ofthe first-stage masking filter H_(mc) ⁽¹⁾(z) is equal to ω_(s); a way ofdetermining the passband edge ω_(pmc) ⁽¹⁾ of the first-stage maskingfilter is as follows: if t₃≥t₄, then$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{3} & {t_{3} \notin {R_{stop}(k)}} \\{\max\left( {\frac{{2\pi\;\left( {k_{3} - 1} \right)} + \varphi_{a}}{M},t_{4}} \right)} & {t_{3} \in {R_{stop}\left( k_{3} \right)}}\end{matrix};} \right.$ if t₃<t₄, then$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{4} & {t_{4} \notin {R_{pass}(k)}} \\{\max\left( {\frac{{2\pi\; k_{4}} - \theta_{a}}{M},t_{3}} \right)} & {t_{4} \in {R_{pass}\left( k_{4} \right)}}\end{matrix};} \right.$ (a) when H_(ma) ⁽²⁾(z^(P)) is used for maskingunnecessary band,t ₃=(2πp−ω _(pma) ⁽²⁾)/P; (b) when the complement of H_(ma) ⁽²⁾(z^(P))is used for masking the unnecessary band,t ₃=(2π(p−1)+ω_(sma) ⁽²⁾)/P; (c) when H_(mc) ⁽²⁾(z^(Q)) is used formasking unnecessary band,t ₄=(2πq−ω _(pmc) ⁽²⁾)/Q; (d) when the complement H_(mc) ⁽²⁾(z^(Q)) isused for masking unnecessary band,t ₄=(2π(q−1)+ω_(smc) ⁽²⁾)/Q; (2) when the transition band of the wholefilter is provided by the complement of H_(a) ⁽²⁾(z^(M)), i.e., in theCase B: the stopband edge ω_(smc) ⁽¹⁾ of the first-stage masking filterH_(mc) ⁽¹⁾(z) is equal to ω_(s); a way of determining the passband edgeω_(pmc) ⁽¹⁾ of the first-stage masking filter H_(mc) ⁽¹⁾(z) is asfollows: if t₃≥t₄, then $\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{3} & {t_{3} \notin {R_{stop}(k)}} \\{\max\left( {\frac{{2\pi\;\left( {k_{3} - 1} \right)} + \varphi_{a}}{M},t_{4}} \right)} & {t_{3} \in {R_{stop}\left( k_{3} \right)}}\end{matrix};} \right.$ if t₃<t₄, then$\omega_{pmc}^{(1)} = \left\{ {\begin{matrix}t_{4} & {t_{4} \notin {R_{pass}(k)}} \\{\max\left( {\frac{{2\pi\; k_{4}} - \theta_{a}}{M},t_{3}} \right)} & {t_{4} \in {R_{pass}\left( k_{4} \right)}}\end{matrix};} \right.$ (a) when H_(mc) ⁽²⁾(z^(Q)) is used for maskingunnecessary band,t ₃=(2π(q−1)+ω_(smc) ⁽²⁾)/Q; (b) when the complement of H_(mc)⁽²⁾(z^(Q)) is used for masking the unnecessary band,t ₃=(2π(q−1)+ω_(smc) ⁽²⁾)/Q; (c) when H_(ma) ⁽²⁾(z^(P)) is used formasking unnecessary band, (d) when the complement H_(ma) ⁽²⁾(z^(P)) isused for masking unnecessary band,t ₄=(2π(p−1)+ω_(sma) ⁽²⁾)/P; wherein t₃ and t₄ are left passband edgesof the passband including the transition band and a first passband onthe left of the transition band, respectively; k₃ is an integersatisfying t₃∈R_(stop)(k₃); k₄ is an integer satisfying t₄∈R_(pass)(k₄);P, Q, M are interpolation factors; θ_(a) is a passband edge of H_(a)⁽²⁾(z); H_(ma) ⁽¹⁾(z)φ_(a) is a stopband edge of H_(a) ⁽²⁾(z).